Abstract
The ligand field is expressed as a linear combination of the full space of operators acting upon a d space of functions. Several orthonormal bases for this operator space are discussed: atomic bases, consisting of irreducible tensor operators of the three-dimensional rotation group R3; molecular bases, consisting of basic symmetry operators, basic energy operators, irreducible tensor operators of the octahedral rotation group 0, or finally, the recommended basis, irreducible tensor operators of all the non-commutative groups of the particular hierarchy of groups chosen. These different bases emphasize different aspects of the symmetry of the chemical system. All operators are also specified with respect to sign.
The weighting factors of the traceless operators in the expression for the ligand field are the observable parameters whose values and/or norms are simply related to the energy splittings caused by the different ligand-field components. These have thus become comparable in a new way both within a given metal complex and between different complexes related to each other by symmetry.
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References
C.J. Ballhausen, Introduction to Ligand Field Theory, Mc Graw-Hill, New York 1962.
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Actually, Griffith [3] introduced the concept of “intermediate symmetry” into ligand-field theory in the more special case of an additive field with fixed geometrical parameters.
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A similar choice of parameters had been used also earlier. For references, see [8] p. 107.
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C.J. Ballhausen, ref. [1] p. 104.
Actually, Ballhausen’s basis choice was one involving complex functions which were not symmetry-adapted to D3⊃C2. A consequence of this is that his non-diagonal element of the type \( < {t_2}\left| {\hat{W}\left( {{D_3}} \right)} \right|e > \) has the opposite sign of that obtained when setting up the energy matrix in his parameters using our basis. This has, of course, no physical consequences and is mentioned here only for completeness.
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It is noteworthy that while \( \hat{C}_2^{{\overline y }} \) mixes \( \overline {\pi c} \) with \( \overline {\delta c} \) and \( \overline \pi s \) with \( \overline {\delta s} \), \( \hat{C}_2^{{\overline x }} \) mixes \( \overline {\pi c} \) with \( \overline {\delta s} \) and \( \overline \pi s \) with \( \overline {\delta c} \). This means that only the \( \hat{C}_2^{{\overline y }} \) choice gives a real energy matrix in |ℓm > or | jm > bases since cosine operators are real linear combinations of \( C_m^{\ell } \) operators and 3j symbols are real [14, p. 291].
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In [20], it has apparently been forgotten [30, Table A 17] that {t1x t1y} and {t2ξ-t2η} span D4 with the same matrices. However, the high priority given to the requirement that Ĉ3 (111) of XYZ obtain a permutation matrix has here only the phase consequence (for a given t1 and t2 set) that \( < {t_1}x\;\left| {\hat{V}\left( {{D_4}} \right)\;} \right|\;{t_2}\xi \; > = - < \;{t_1}y\;\left| {\hat{V}\left( {{D_4}} \right)\;} \right|\;{t_2}\eta > \) In the trigonal case of [20], this priority has more consequence. Here Griffith’s basis functions for t1 and t2 are {t1ρ t1σ t1τ} = {t1b-t1a t1c}, {t2ρ t2σ t2τ} = {-t2α-t2β t2γ}, given in terms of those of (69), so that they are adapted to 0 ⊃ (C3) (where the parentheses here refer to the fact that the group C3 is only represented by the “most reduced” representations consistent with a real basis for e(C3) [14, p. 235]) rather than 0 ⊃ D3 as claimed [20, p. 19]. Using the above equations the rather special consequences for matrix elements can be readily seen.
J.S. Griffith, The Theory of the Transition-Metal Ions. Cambridge Univ. Press 1961.
In order for u < v to be meaningful some sort of priority order must be defined for the indices. In the particular case ôuv = ôvu and only one of these occurs in the summation.
\( \left| {\hat{W}\left( {{D_3}} \right)} \right| = {\left( { < \hat{W}\left( {{D_3}} \right)\left| {\hat{W}\left( {{D_3}} \right) > } \right.} \right)^{{\frac{1}{2}}}} \) is the norm (or length) of the total observable field. We can also talk about the norm of the rhombo-hedral field which is \( \left| {\hat{V}\left( {{D_3}} \right)} \right| \), relating to a particular operator subspace, which can be expressed either in a molecular or an atomic scheme of the hierarchy (1) \( \left| {\hat{V}\left( {{D_3}} \right)} \right| = \sqrt {{{{\left( {N_{\gamma }^{{{t_2}{t_2}\left| {{t_2}} \right.}}} \right)}^2} + {{\left( {N_{\gamma }^{{{t_2}}}\left( {{t_2}e} \right)} \right)}^2} = \sqrt {{{{\left( {N_{{{t_2}\gamma }}^d} \right)}^2} + {{\left( {N_{{{t_2}\gamma }}^g} \right)}^2}}} }} \)
Alternatively, the expression (61) can be used to solve the linear equations \( N_{{a1l}}^g = < \hat{W}\left( {{D_3}} \right)\left| {\hat{N}_{{a1l}}^g} \right. > \) ;\( N_{{{t_2}\gamma }}^g = < \hat{W}\left( {{D_3}} \right)\left| {\hat{N}_{{{t_2}\gamma }}^g} \right. > \) for \( M_{{a1l}}^g \) and \( M_{{\overline \delta }}^g \) (see also [6, footnote 28]).
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© 1987 D. Reidel Punliashing Company, Dordrecht, Holland
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Schäffer, C. (1987). The Orthonormal Operator Formulation of Symmetry-Based Ligand Fields: Rhombohedral Hierarchies as a General Example. In: Avery, J., Dahl, J.P., Hansen, A.E. (eds) Understanding Molecular Properties. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3781-9_8
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DOI: https://doi.org/10.1007/978-94-009-3781-9_8
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